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Primoz Moravec

Publications and source records attributed to Primoz Moravec.

At least 19 recordsLinked to original sources
arXiv2026

Uniform exponent bounds for integral group homology

We prove that, in each fixed degree, the exponent of the integral homology of a finite group is bounded solely in terms of the degree and the exponent of the group. The proof combines the solution of the restricted Burnside problem with a representability property of the bar construction and may be viewed as a torsion analogue of the method of acyclic models. We also use the Lyndon--Hochschild--Serre spectral sequence to obtain explicit bounds for finite solvable and nilpotent groups in terms of their derived length and nilpotency class.

math.GR↗
arXiv2026

Polynomial permutation stability, soficity, and universal polynomial groups

We introduce polynomial permutation stability, extending classical permutation stability from homomorphisms to polynomial maps of groups. The universal polynomial group $\operatorname{Pol}_s(G)$ provides a natural framework for this theory, and we prove that polynomial stability of degree $s$ is equivalent to permutation stability of $\operatorname{Pol}_s(G)$. This yields, in particular, polynomial stability of all finite groups in degree two. We develop a corresponding theory of polynomial sofic approximations and show that, for countable groups, it does not give a new notion of soficity. Nevertheless, the associated universal polynomial groups contain new structural information: in degree two they embed into a wreath product, which allows us to characterize their soficity in terms of the original group. We also formulate and study weak polynomial stability through the weak stability theory of universal polynomial groups. For finite groups, we investigate the structure of higher-degree universal polynomial groups. We obtain general obstructions to amenability and prove largeness results for a family of finite groups with perfect derived subgroup and abelianization of order two.

math.GR↗
arXiv2025

Absence of twisting for non-trivial discrete torsion

We study discrete torsion for the $n$--torus with finite symmetry group $G$ from the Dijkgraaf--Witten viewpoint. A class in $H^n(G,U(1))$ assigns a phase to each flat $G$--bundle, equivalently to each commuting $n$--tuple in $G$ up to conjugation. We introduce the subgroup $\Br^n(G)\subseteq H^n(G,U(1))$ of \emph{untwisted} classes, those whose Dijkgraaf--Witten phases are trivial on all commuting tuples, and derive a universal coefficient exact sequence involving this invariant. In degree $2$ this recovers the Bogomolov multiplier / unramified Brauer group. We implement algorithms for computing $\Br^n(G)$ and corresponding torus partition functions, and report on computations for families of finite subgroups of $\SU(4)$.

math.GR↗
arXiv2025

Quasi-affine and quasi-quadratic maps of groups with non-abelian targets

It is shown that the middle quasi-homomorphisms of Fujiwara and Kapovich are precisely constant perturbations of quasi-homomorphisms. Quasi-polynomial maps are defined and their constructibility is explored. In particular, it is shown that a large class of quasi-quadratic maps into torsion-free hyperbolic groups is rigid with respect to bounded perturbations.

math.GR↗
arXiv2024

The powerful class of Sylow subgroups of finite groups

The paper explores the effect of powerful class of Sylow $p$-subgroups of a given finite group on control of transfer or fusion. We also find an explicit bound for the $p$-length of a $p$-solvable group in terms of the poweful class of a Sylow $p$-subgroup.

math.GR↗
arXiv2023

The powerful class of groups

Pro-$p$ groups of finite powerful class are studied. We prove that these are $p$-adic analytic, and further describe their structure when their powerful class is small. It is also shown that there are only finitely many finite $p$-groups of fixed coclass and powerful class.

math.GR↗
arXiv2019

On two group functors extending Schur multipliers

Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.

math.GR↗
arXiv2018

On finite $p$-groups satisfying given laws

A variety of groups does not contain all metabelian groups if and only if there is an absolute bound for the nilpotency classes of powerful $p$-groups in the given variety. Similarly, a variety contains only finitely many finite $p$-groups of any given coclass if and only if not every group that is an extension of an abelian group by an elementary abelian $p$-group belongs to that variety.

math.GR↗
arXiv2018

Exponents of Bogomolov multipliers

We prove that if $G$ is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of $G$ in the following four cases: (i) $G$ is metabelian, (ii) $\exp G=4$, (iii) $G$ is nilpotent of class $\le 5$, or (iv) $G$ is a $4$-Engel group.

math.GR↗
arXiv2016

Groups in which every non-abelian subgroup is self-normalized

We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above property. We also describe all infinite soluble groups in this class.

math.GR↗
arXiv2015

Groups in which every non-abelian subgroup is self-centralizing

We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups with the above property.

math.GR↗
arXiv2015

Commutativity preserving extensions of groups

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrising such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.

math.GR↗
arXiv2013

Universal commutator relations, Bogomolov multipliers, and commuting probability

Let $G$ be a finite $p$-group. We prove that whenever the commuting probability of $G$ is greater than $(2p^2 + p - 2)/p^5$, the unramified Brauer group of the field of $G$-invariant functions is trivial. Equivalently, all relations between commutators in $G$ are consequences of some universal ones. The bound is best possible, and gives a global lower bound of $1/4$ for all finite groups. The result is attained by describing the structure of groups whose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all of their proper subgroups and quotients are trivial. Applications include a classification of $p$-groups of minimal order that have nontrivial Bogomolov multipliers and are of nilpotency class $2$, a nonprobabilistic criterion for the vanishing of the Bogomolov multiplier, and establishing a sequence of Bogomolov's absolute $γ$-minimal factors which are $2$-groups of arbitrarily large nilpotency class, thus providing counterexamples to some of Bogomolov's claims. In relation to this, we fill a gap in the proof of triviality of Bogomolov multipliers of finite simple groups.

math.GR↗
arXiv2012

Unramified Brauer groups of finite and infinite groups

The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a Hopf-type formula, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. A new description of the Bogomolov multiplier of a nilpotent group of class two is obtained. We define the Bogomolov multiplier within K-theory and show that proving its triviality is equivalent to solving a long-standing problem posed by Bass. An algorithm for computing the Bogomolov multiplier is developed.

math.GR↗